A Smarandache Strong-Weak Structure on a set S means a structure on S that has two proper subsets: P with a stronger structure, and Q with a weaker structure.

By proper subset of a set S, we mean a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any.

   In any field, a Smarandache strong-weak n-structure on a set S means a structure {w₀} on S such that there exist two chains of proper subsets P_n-1 < P_n-2 < … < P₂ < P₁ < S and Q_n-1 < Q_n-2 < … < Q₂ < Q₁ < S, where '<' means 'included in', whose corresponding stronger structures verify the chain {w_n-1} > {w_n-2} > … > {w₂} > {w₁} > {w₀} and respectively the weaker structures verify the chain {v_n-1} < {v_n-2} < … < {v₂} < {v₁} < {v₀}, where '>' signifies 'strictly stronger' (i.e. structure satisfying more axioms) and '<' signifies 'strictly weaker' (i.e. structure satisfying less axioms).

   And by structure on S we mean a structure {w} on S under the given operation(s).

   As a particular case, a Smarandache strong-weak 2-structure (two levels only of structures in algebra) on a set S, is a structure {w₀} on S such that there exist two proper subsets P and Q of S, where P is embedded with a stronger structure than {w₀}, while Q is embedded with a weaker structure than {w₀}.

For example, a Smarandache strong-weak monoid is a monoid that has a proper subset which is a group, and another proper set which is a semigroup.

Also, a Smarandache strong-weak ring is a ring that has a proper subset which is a field, and another proper subset which is a near-ring.  

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